Optimal beamforming with combined channel and path CSIT for multi-cell multi-user MIMO

Optimal Beamforming with Combined Channel and Path CSIT for Multi-cell Multi-User MIMO

Wassim Tabikh^‡¶, Dirk Slock^‡, Yi Yuan-Wu^¶

‡EURECOM, Sophia-Antipolis, France, Email: tabikh@eurecom.fr, slock@eurecom.fr

¶Orange Labs, Issy-les-Moulineaux, France, Email: yi.yuan@orange.com

Abstract—We consider a combined form of partial CSIT (Channel State Information at the Transmitter(s) (Tx)), compris- ing both channel estimates (mean CSIT) and covariance CSIT. In particular multipath induced structured low rank covariances are considered that arise in Massive MIMO and mmWave settings.

For the beamforming optimization, we first revisit Weighted Sum Rate (WSR) maximization with perfect CSIT and introduce yet another equivalent approach: Weighted Sum Unbiased MSE (WSUMSE). We then turn to the partial CSIT case where we consider Expected WSR (EWSR) maximization for which EWSUMSE turns out to be a better approximation compared to the existing EWSMSE approach. These approaches also allow an uplink/downlink duality interpretation for partial CSI.

I. INTRODUCTION

In this paper, Tx may denote transmit/transmitter/

transmission and Rx may denote receive/receiver/reception.

Interference is the main limiting factor in wireless transmis- sion. Base stations (BSs) disposing of multiple antennas are able to serve multiple Mobile Terminals (MTs) simultaneously, which is called Spatial Division Multiple Access (SDMA) or Multi-User (MU) MIMO. However, MU systems have precise requirements for Channel State Information at the Tx (CSIT) which is more difficult to acquire than CSI at the Rx (CSIR).

Hence we focus here on the more challenging downlink (DL).

The main difficulty in realizing linear IA for MIMO I(B)C is that the design of any BS Tx filter depends on all Rx filters whereas in turn each Rx filter depends on all Tx filters [1].

As a result, all Tx/Rx filters are globally coupled and their design requires global CSIT. To carry out this Tx/Rx design in a distributed fashion, global CSIT is required at all BS [2]. The overhead required for this global distributed CSIT is substantial, even if done optimally, leading to substantially reduced Net Degrees of Freedom (DoF) [3].

The recent development of Massive MIMO (MaMIMO) [4]

opens new possibilities for increased system capacity while at the same time simplifying system design. We refer to [5] for a further discussion of the state of the art, in which MIMO IA requires global MIMO channel CSIT. Recent works focus on intercell exchange of only scalar quantities, at fast fading rate, as also on two-stage approaches in which the intercell interference gets zero-forced (ZF). Also, massive MIMO in most works refers actually to MU MISO.

Whereas the exploitation of covariance CSIT may be ben- eficial, in a MaMIMO context it may quickly lead to high computational complexity and estimation accuracy issues.

Computational complexity may be reduced (and the benefit of

covariance CSIT enhanced) in the case of low rank or related covariance structure, but the use and tracking of subspaces may still be cumbersome. In the pathwise approach, these subspaces are very parsimoniously parameterized. In a FDD setting, these parameters may even be estimated from the uplink (UL). In a TDD setting with reciprocity, the channel estimation error may account for time variation also in the UL/DL ping-pong. As opposed to the instantaneous channel CSIT, the path CSIT is not affected by fast fading.

Whereas path CSIT by itself may allow zero forcing (ZF) [6], which is of interest at high SNR, we are particularly concerned here with maximum Weighted Sum Rate (WSR) designs accounting for finite SNR. ZF of all interfering links leads to significant reduction of useful signal strength. Massive MIMO makes the pathwise approach viable: the (cross-link) beamformers (BF) can be updated at a reduced (slow fading) rate, parsimonious channel representation facilitates not only uplink but especially downlink channel estimation, the cross- link BF can be used to significantly improve the downlink direct link channel estimates (in FDD), minimal feedback can be introduced to perform meaningful WSR optimization at a finite SNR (whereas ZF requires much less coordination).

In this paper we first introduce a new approach to max- imizing WSR, called minimum Weighted Sum Unbiased MSE (WSUMSE). In the perfect CSIT case, WSUMSE also converges to a local optimum of WSR. We then consider various approaches for maximizing Expected WSR (EWSR) for the case of partial CSIT. The existing EWSMSE approach improves over Naive EWSR (NEWSR) by accounting for covariance CSIT in the interference. This can have significant impact, even on the sumrate prelog (DoF) if the instantaneous channel CSIT quality does not scale with SNR. A further improvement is proposed here in the EWSUMSE approach which represents a better approximation of the EWSR. In a MaMIMO setting, the way mean and covariance CSIT are combined in the EWSMSE or WSUMSE approaches for the interference terms becomes equally optimal as in the EWSR for a large number of users. EWSUMSE represents an improvement over EWSMSE for capturing the signal power (matched filtering and diversity aspects) and only leads to a finite (dB) gain in ESINR, but its remaining approximation er- ror over EWSR may be limited. Strictly speaking, in the large number of users setting, EWSMSE≤EWSR≤EWSUMSE.

The step from EWSMSE to EWSUMSE also deals with the following question. Covariance CSIT can be used to improve

the channel estimate from a basic deterministic estimate to a Bayesian estimate. The question then arises: is that enough?

The answer is no and a first take at this issue is proposed here.

This paper is a followup on [7] from which we reproduce some sections to ease reading. In [7] we introduced a heuristic to design the Tx separately using path CSIT only. It turns out that this heuristic is recovered by the EWSUMSE approach proposed here, which furthermore provides expressions for a number of auxiliary quantities that are needed and allows the combination of channel estimate and path CSIT.

II. CHANNEL(INFORMATION) MODELS

In this section we drop the user index kfor simplicity.

A. Specular Wireless MIMO Channel Model

The MIMO channel transfer matrix at any particular sub- carrier of a given OFDM symbol can be written as [8], [9]

H=

N_p

X

i=1

Aie^jψihr(φi)h^T_t(θi) =B A^H (1) where there are Np (specular) pathwise contributions with

• Ai>0: path amplitude

• θi: direction of departure (AoD)

• φi: direction of arrival (AoA)

• ht(.),hr(.):M/N×1 Tx/Rx antenna array response with||ht(.)||= 1,||hr(.)||=N, and

B= [h_r(φ₁)h_r(φ₁)· · ·]





 e^jψ1

e^jψ2 . ..





,A^H=





 A₁

A2

. ..











 h^T_t(θ₁) h^T_t(θ2)

...





 (2) The antenna array responses are just functions of angles AoD, AoA in the case of standard antenna arrays with scatterers in the far field. In the case of distributed antenna systems, the array responses become a function of all position parameters of the path scatterers. The fast variation of the phasesψ_i(due to Doppler) and possibly the variation of the A_i (when the nominal path represents in fact a superposition of paths with similar parameters) correspond to the fast fading. All the other parameters vary on a slower time scale and correspond to slow fading.

B. Dominant Paths Partial CSIT Channel Model

Assuming the Tx disposes of not much more than the information about r dominant path AoDs, we shall consider the following MIMO (Ricean) channel model

H=B A^H(θ) +p

βHe⁰ (3)

which follows from (1), (2) except restricted to therstrongest paths, with the rest modeled by √

βHe⁰ (elements i.i.d. ∼ CN(0, β), independent of the ψi). Averaging of the path phases ψi, we get for the Tx side covariance matrix

EH^HH=NC_t=N(AA^H+β I_M) (4) since due to the normalization of the antenna array responses, EB^HB=diag{[hr(φ1)hr(φ2)· · ·]^H[hr(φ1)hr(φ2)· · ·]} =

NI. Note that the pathwise channel model, which leads here to a type of Tx covariance CSIT, does not lead to the usual separable covariance case, which is discussed e.g. [5].

C. Combined Channel and Path Partial CSIT

Now, the Tx dispose also of a (det erministic) channel estimate

Hb =H+ 1

√

NHedC^1/2_d (5) where the elements of He_d are i.i.d. ∼ CN(0,1), and typically C_d = σ²

heI_M. The combination of the channel estimate with the prior information leads to the (posterior) LMMSE estimate. The non-separable prior covariance should in principle be accounted for properly in the LMMSE estima- tion. However, to simplify beamformer design, we shall force a separable prior model by considering all ofBto be unknown with elements that are i.i.d. ∼ CN(0,1). As a result we can interpretHto be of the formH=H Ce ^1/2_t and we get for the LMMSE estimate

bb

H=Hbd(Ct+Cd)⁻¹Ct=H+HepC^1/2p

Cp=Cd(Ct+Cd)⁻¹Ct

(6) where Hbb and Hep are independent. Note that we get for the MMSE estimate of a quadratic quantity of the form

E_H|bH

dH^HH= b Hb

H

bb

H+NCp=W. (7) Let us emphasize that this MMSE estimate implies W = arg minT E_H|bH

d||H^HH−T||². It averages out to EHb_dW= E_H,bH

dH^HH= EHH^HH=NCt. (8) III. STREAMWISEIBC SIGNALMODEL

In this paper we shall consider mostly a per stream approach (which in the perfect CSI case would be equivalent to per user). In an IBC formulation, one stream per user can be expected to be the usual scenario. In the development below, in the case of more than one stream per user, treat each stream as an individual user. So, consider again an IBC withCcells with a total ofKusers. We shall consider a system-wide numbering of the users. Userkis served by BSbk. TheNk×1 received signal at userk in cellbk is

yk=Hk,b_kgkxk

| {z } signal

+X

i6=k

b_i=b_k

Hk,b_kgixi

| {z } intracell interf.

+X

j6=bk

X

i:bi=j

Hk,jgixi

| {z }

intercell interf.

+vk

(9) wherex_k is the intended (white, unit variance) scalar signal stream,H_k,bk is the N_k ×M_bk channel from BSb_k to user k. BS b_k serves K_bk = P

i:bi=bk1 users. We considering a noise whitened signal representation so that we get for the noise vk ∼ CN(0, IN_k). The Mb_k ×1 spatial Tx filter or beamformer (BF) isgk. Treating interference as noise, userk will apply a linear Rx filterfk to maximize the signal power (diversity) while reducing any residual interference that would

not have been (sufficiently) suppressed by the BS Tx. The Rx filter output is bx_k =f_k^Hy_k

bxk = f_k^HHk,b_kgkxk+

K

X

i=1,6=k

f_k^HHk,b_igixi+f_k^Hvk

= f_k^Hhk,kxk+X

i6=k

f_k^Hhk,ixi+f_k^Hvk

(10)

where hk,i = Hk,b_igi is the channel-Tx cascade vector.

ZF (IA) feasibility for both the general reduced rank MIMO channels case and the pathwise MIMO case has been discussed in [6], in particular also when only based on Tx side covariance CSIT. Also the role of Rx antennas is highlighted and a comparison with FIR ZF in an asynchronous scenario is presented.

IV. MAXWSRWITHPERFECTCSIT

Consider as a starting point for the optimization the weighted sum rate (WSR)

W SR=W SR(g) =

K

X

k=1

u_k ln 1 ek

(11) where grepresents the collection of BFs gk, theuk are rate weights, the ek = ek(g) are the Minimum Mean Squared Errors (MMSEs) for estimating the xk:

1 ek

=1+g^H_k H^H_k,b

kR⁻¹

k Hk,b_kgk= (1−g^H_k H^H_k,b

kR⁻¹_k Hk,b_kgk)⁻¹ R_k=H_k,bkQ_kH^H_k,b

k+R_k, Q_i=g_ig^H_i , R_k=X

i6=k

Hk,b_iQiH^H_k,bi+IN_k.

(12) R_k,R_kare the total and interference plus noise Rx covariance matrices resp. and ek is the MMSE obtained at the output xbk=f_k^Hyk of the optimal (MMSE) linear Rxfk,

fk =R⁻¹_k Hk,b_kgk =R⁻¹_k hk,k. (13) The WSR cost function needs to be augmented with the power constraints

X

k:b_k=j

tr{Qk} ≤Pj. (14) A. From Max WSR to Min WSMSE

For a general Rx filterfk we have the MSE ek(fk,g) = (1−f_k^HHk,b_kgk)(1−g^H_kH^H_k,b

kfk) +P

i6=kf_k^HHk,b_igig^H_i H^H_k,b

ifk+||fk||²= 1−f_k^HHk,b_kgk

−g^H_kH^H_k,b

kf_k+X

i

f_k^HH_k,big_ig^H_i H^H_k,b

if_k+||f_k||².

(15) TheW SR(g)is a non-convex and complicated function ofg.

Inspired by [10], we introduced [11], [1] an augmented cost function, the Weighted Sum MSE,W SM SE(g,f, w)

=

K

X

k=1

uk(wk ek(fk,g)−lnwk) +

C

X

i=1

λi( X

k:bk=i

||gk||²−Pi) (16)

where λi = Lagrange multipliers. After optimizing over the aggregate auxiliary Rx filters f and weights w, we get the WSR back:

min

f,w W SM SE(g,f, w) =−W SR(g) +

K

X

k=1

u_k (17) The advantage of the augmented cost function: alternating optimization leads to solving simple quadratic or convex functions:

minwk

W SM SE ⇒ wk= 1/ek

min

f_k W SM SE ⇒f_k= (X

i

H_k,big_ig^H_i H^H_k,b

i+I_Nk)⁻¹H_k,bkg_k ming_k W SM SE ⇒

gk= (P

iuiwiH^H_i,b

kfif_i^HHi,b_k+λb_kIM)⁻¹H^H_k,b

kfkukwk (18) UL/DL duality: the optimal Tx filter gk is of the form of a MMSE linear Rx for the dual UL in whichλplays the role of Rx noise variance andukwk plays the role of stream variance.

B. Difference of Convex Functions Programming

In a classical difference of convex functions (DC pro- gramming) approach, Kim and Giannakis [12] propose to keep the concave signal terms and to replace the convex interference terms by the linear (and hence concave) tangent approximation. More specifically, consider the dependence of WSR onQk alone. Then

W SR=ukln det(R⁻¹

k Rk) +W SR_k, W SR_k=PK

i=1,6=kuiln det(R⁻¹

i Ri) (19)

whereln det(R⁻¹

k R_k)is concave inQ_kandW SR_k is convex inQ_k. Since a linear function is simultaneously convex and concave, consider the first order Taylor series expansion inQ_k around Q⁰ (i.e. allQ⁰_i) with e.g.R⁰_i=R_i(Q⁰), then W SR_k(Qk,Q⁰)≈W SR_k(Q⁰_k,Q⁰)−tr{(Qk−Q⁰_k)T⁰_k} T⁰

k =− ∂W SR_k(Q_k,Q⁰)

∂Qk

_Q0

k,Q⁰

=

K

X

i6=k

uiH^H_i,bk(R⁰⁻¹

i −R⁰_i⁻¹)Hi,b_k

(20) Note that the linearized (tangent) expression for W SR_k constitutes a lower bound for it. Now, dropping constant terms, reparameterizing the Qk = gkg^H_k, performing this linearization for all users, and augmenting the WSR cost function with the constraints, we get the Lagrangian

W SR(g,g⁰, λ) =

C

X

j=1

λjPj+

K

X

k=1

u_kln(1 +g^H_k S⁰_kg_k)−g_k^H(T⁰_k+λ_bkI)g_k

(21)

where

S⁰_k=H^H_k,bkR⁰⁻¹

k Hk,b_k. (22)

The gradient (w.r.t.gk) of this concave WSR lower bound is actually still the same as that of the original WSR criterion!

And it allows an interpretation as a generalized eigenvector condition

S⁰_kgk= 1 +g^H_k S⁰_kgk

u_k (T⁰_k+λb_kI)gk (23) or hence g_k = Vmax(S⁰_k,T⁰

k +λb_kI) is the (normalized)

”max” generalized eigenvector of the two indicated matrices, with max eigenvalue σk =σmax(S⁰_k,T⁰

k+λb_kI). Letσ⁽¹⁾_k = g^H_k S⁰_kg_k,σ⁽²⁾_k =g^H_k T⁰

kg_k. The advantage of formulation (21) is that it allows straightforward power adaptation: introducing stream powers pk ≥0 and substitutinggk =√

pkg_k in (21) yields

W SR=

C

X

j

λjPj+

K

X

k=1

{ukln(1 +pkσ⁽¹⁾_k )−pk(σ_k⁽²⁾+λb_k)}

(24) which leads to the following interference leakage aware water filling (WF)

p_k = u_k σ⁽²⁾_k +λ_bk

− 1 σ_k⁽¹⁾

!⁺

(25) where the Lagrange multipliers are adjusted to satisfy the power constraints P

k:b_k=jp_k = P_j. This can be done by bisection and gets executed per BS. Note that some Lagrange multipliers could be zero. Note also that as with any alternating optimization procedure, there are many updating schedules possible, with different impact on convergence speed. The quantities to be updated are theg_k, thepk and the λl.

Note that the DC programming approach, which avoids introducing Rxs, can at every BF update allow to introduce an arbitrary number of streams per user by determining mul- tiple dominant generalized eigenvectors, and then let the WF operation decide how many streams can actually be sustained.

V. FROMMAXWSRTOMINWEIGHTEDSUMUNBIASED

MSE (WSUMSE)

For the Rx output xb_k to be an unbiased estimator for the Tx signalx_k, we require

E_|xkxb_k=x_k ⇒ f_k^HH_k,bkg_k = 1. (26) If the Tx/Rx filters satisfy the unbiasedness constraint, then we get the Unbiased MSE (UMSE)

e^u_k(fk,g) =X

i6=k

f_k^HHk,b_igig^H_i H^H_k,bifk+||fk||². (27) In the complex case, it is more convenient to work with the alternative unbiasedness constraint

|f_k^HHk,b_kgk|²= 1 (28) in which the phase uncertainty does not affect SINR or Gaussian capacity. Now consider the following

augmented cost function, the Weighted Sum UMSE, W SU M SE(g,f, w₁, w₂)

=

K

X

k=1

u_k(w_1,ke^u_k(f_k,g)−lnw_1,k−w_2,k(e^u_k(f_k,g)+1)+lnw_2,k) +

K

X

k=1

µ_k(1− |f_k^HH_k,bkg_k|²) +

C

X

i=1

λ_i( X

k:b_k=i

||g_k||²−P_i) (29) where the λ_i,µ_k are Lagrange multipliers. After optimizing over the aggregate auxiliary Rx filtersf and weights w₁,w₂, we get the WSR back:

min

f min

w₁ max

w₂ W SU M SE(g,f, w1, w2) =−W SR(g) (30) where W SR = P

kukln(1 + _e¹u

k). The augmented cost function can again be conveniently optimized by alternating optimization:

minw_1,kmax

w_2,kW SU M SE ⇒ w1,k= 1/e^u_k, w2,k = 1/(e^u_k+ 1) min

f_k W SU M SE⇒f_k=R⁻¹

k H_k,bkg_kµkg^H_k H^H_k,b

kfk

u_kw_k mingk

W SU M SE ⇒ gk= (T_k+λb_kIM)⁻¹H^H_k,b

kfkµkf_k^HHk,b_kgk (31) where wk =w1,k−w2,k >0. Note that|f_k^HHk,b_kgk| = 1.

We can choose the phase such that f_k^HH_k,bkg_k = 1. If we also reparameterizeµk =ukwkµ⁰_k, where the µ⁰_k are chosen to satisfyf_k^HHk,b_kgk = 1, then we can rewrite

fk=R⁻¹

k Hk,b_kgkµ⁰_k,gk= (T_k+λb_kIM)⁻¹H^H_k,b

kfkukwkµ⁰_k which are proportional to the Tx/Rx found by the WSMSE approach.

A word about convergence. Note that given thef andg(or in other words thee^u_k), the WSUMSE decomposes into thew₁ andw₂dependencies, which can be optimized in parallel. The optimization over w₁, w₂ jointly decreases the cost function sinceln(_1+e^euku

k

)is an increasing function ofe^u_k. Now, as long as the optimization over the w1, w2 is done jointly, every alternating optimization in (31) decreases the WSUMSE cost function.

VI. EXPECTEDWSR (EWSR)

For the WSR criterion, we have assumed so far that the channel H is known. The scenario of interest however is that of partial CSIT. Once the CSIT is imperfect, various optimization criteria could be considered, such as outage capacity. Here we shall consider the expected weighted sum rate E

H|HbbW SR(g,H) = EW SR(g) = E

H|Hbb

X

k

u_kln(1 +g^H_k H^H_k,b

kR⁻¹

k H_k,bkg_k) (32) where we now underlign the dependence of various quantities on H. The EWSR in (32) corresponds to perfect CSIR since the optimal Rx filters fk as a function of the ag- gregate H have been substituted, namely W SR(g,H) =

maxfP

kuk(−ln(ek(fk,g))). At high SNR, max EWSR at- tempts ZF. Now we consider various deterministic approxima- tions for the EWSR.

VII. EWSR LOWERBOUND: EWSMSE

The criterion EW SR(g) is difficult to compute and to maximize directly. It is much more attractive to consider EH|Hbbe_k(f_k,g,H) as in [13] since e_k(f_k,g,H) is quadratic inH. Hence consider optimizing the expected weighted sum MSE E

H|HbbW SM SE(g,f, w,H).

minf,w E

H|HbbW SM SE(g,f, w,H)

≥ E

H|Hbbmin_f,wW SM SE(g,f, w,H) =−EW SR(g) (33) or hence

EW SR(g)≥ −min

f,w E

H|HbbW SM SE(g,f, w,H). (34) So now only a lower bound to the EWSR gets maximized, which corresponds in fact to theCSIR being equally partial as the CSIT (whereas the EWSR criterion corresponds to partial CSIT but perfect CSIR). Now, since

EH|HbbHQH^H =HQbb Hbb

H

+tr{QCp}IN

EH|HbbH^HQH=Hbb

H

QH+bb tr{Q}Cp

(35)

we get E

H|Hbbek = bbek = 1−2<{f_k^Hb

Hbk,b_kgk}+PK i=1f_k^Hb

Hbk,b_igig^H_i b Hb

H k,bifk

+||fk||²PK

i=1g^H_i C_p,k,big_i+||fk||².

(36) whereCt,k,b_i are Tx side (LMMSE error) covariance matrices of H_k,bi. Note that the signal term disappears if Hbb_k = 0 ! Hence the EWSMSE lower bound is (very) loose unless the Rice factor is high, and is useless in the absence of channel estimates. Alternating optimization as before leads to

minwk

EW SM SE ⇒ wk= 1/bbek

min

fk

EW SM SE ⇒fk=b Rb

−1 k b

Hbk,b_kgk

ming_k EW SM SE ⇒gk= (Tbbk+λb_kIM)⁻¹Hbb

H

k,bkfkukwk

(37) where

bb Rk =P

iHbbk,b_igig^H_i Hbb

H

k,bi+ (1 +P

ig^H_i Cp,k,b_igi)IN_k

bb

T_k =PK

i=1u_iw_i(Hbb

H

i,b_kf_if_i^HHbb_i,bk+||f_i||²C_p,k,bi). (38) VIII. EXPECTEDWEIGHTEDSUMUNBIASEDMSE

(EWSUMSE)

Consider now the Expected WSUMSE (EWSUMSE) cri- terion obtained by averaging (29) over E

H|Hbb. Alternating

optimization leads to wk = 1/(bbe^u_k(bbe^u_k+ 1)) fk=Vmax(b

Hbk,b_kgkg^H_k b Hb

H

k,bk+g_k^HCp,k,b_kgkIN_k,b Rb_k) gk =Vmax(b

Hb

H

k,bkfkf_k^Hb

Hbk,b_k+||fk||²Cp,k,b_k,b

Tb_k+λb_kIM) (39) where Rbb_k and Tbb_k are like Rbb_k and Tbb_k in (38) but without the term for userk. Note that in case of no channel estimate,

bb

H_k,bk= 0, then

f_k =V_min(Rbb_k)

gk=Vmax(Cp,k,b_k,Tbb_k+λb_kIM).

(40) The Rx and Tx filters optimizing the EWSUMSE criterion also optimize the ESEINR (Expected Signal to Expected Interfer- ence plus Noise Ratio) and (optimally weighted) ESELNR (Expected Signal to Expected Leakage plus Noise Ratio):

ESEINRk =f_k^H(b

Hbk,b_kgkg^H_k b Hb

H

k,bk+g_k^HCp,k,b_kgkIN_k)fk

f_k^Hb Rb_kfk

ESELNRk =g^H_k(b Hb

H

k,bkfkf_k^Hb

Hbk,b_k+||fk||²Cp,k,b_k)gk

g^H_k (b

Tb_k+λb_kIM)gk

. (41) Of course, the scale factors of the Tx and Rx filters in (39) need to be adjusted, which could be done as in [10] or (with the Lagrange multipliers also) using an appropriate adaptation of the DC programming waterfilling. In that case,f_k is left to be unit norm as in (39) whereasg_k from (39) is interpreted as g_k. The stream powers and Lagrange multipliers get adapted from (24),(25) with b

Tb_k replacingT_k and Sk being replaced by

bb S_k=Hbb

H k,b_kRbb

−1

k Hbb_k,bk+tr{Rbb

−1

k }C_p,k,bk. (42) IX. DISCUSSIONEWSR APPROXIMATIONS

In the case of partial CSIT we get for the symbol estimate xbk=f_k^HHbbk,b_kgkxk+f_k^HHek,b_kgkxk

| {z } sig. ch. error +

K

X

i=1,6=k

(f_k^Hb

Hbk,b_igixi+ f_k^HHek,b_igixi

| {z } interf. ch. error

) +f_k^Hvk. (43)

A first approach would be Naive EWSR (NEWSR) which would just replace H by Hbb in a perfect CSIT approach.

It ignores the channel estimation error in both signal and interference terms. The EWSMSE approach improves over NEWSR by accounting for covariance CSIT in the interfer- ence. This can have significant impact, even on the DoF if the instantaneous channel CSIT quality does not scale with SNR. However, note from (38) that EWSMSE also moves the channel estimation error in the signal term to the interference plus noise. A further improvement is proposed here in the

EWSUMSE approach which represents a better approximation of the EWSR. In EWSUMSE, the channel estimation error in the signal term is accounted for in the signal power. In a MaMIMO setting, the way mean and covariance CSIT are combined in the WSUMSE approach for the interference terms becomes equally optimal as in the EWSR for a large number of users. Indeed, in the MaMIMO setting the interference becomes Gaussian due to the Central Limit Theorem and hence is characterized by its covariance. EWSUMSE further- more represents an improvement over EWSMSE for capturing the signal power (matched filtering and diversity aspects) but only leads to a finite (dB) gain in ESEINR, though its remaining approximation error over EWSR may be limited. In the MaMIMO setting, EWSUMSE represents a EWSR upper bound due to the concavity of ln(.).

Both EWSMSE and EWSUMSE lead to UL/DL duality for partial CSIT, but only EWSUMSE treats signal and interfer- ence terms similarly.

ACKNOWLEDGMENTS

EURECOM’s research is partially supported by its industrial members: ORANGE, BMW, SFR, ST Microelectronics, Sy- mantec, SAP, Monaco Telecom, iABG, by the projects ADEL (EU FP7) and DIONISOS (French ANR). The research of Orange Labs is partially supported by the EU H2020 project Fantastic5G.

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